Stabilizability and percolation in the infinite volume sandpile model

Abstract

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $\mu$, are $\mu$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $\mu$ a product measure with density $\rho=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $\mu$ is not stabilizable. Furthermore, we study, for values of $\rho$ such that $\mu$ is stabilizable, percolation of toppled sites. We find that for $\rho>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.